Can We Identify Non-Stationary Dynamics of Trial-to-Trial Variability?

Identifying sources of the apparent variability in non-stationary scenarios is a fundamental problem in many biological data analysis settings. For instance, neurophysiological responses to the same task often vary from each repetition of the same experiment (trial) to the next. The origin and functional role of this observed variability is one of the fundamental questions in neuroscience. The nature of such trial-to-trial dynamics however remains largely elusive to current data analysis approaches. A range of strategies have been proposed in modalities such as electro-encephalography but gaining a fundamental insight into latent sources of trial-to-trial variability in neural recordings is still a major challenge. In this paper, we present a proof-of-concept study to the analysis of trial-to-trial variability dynamics founded on non-autonomous dynamical systems. At this initial stage, we evaluate the capacity of a simple statistic based on the behaviour of trajectories in classification settings, the trajectory coherence, in order to identify trial-to-trial dynamics. First, we derive the conditions leading to observable changes in datasets generated by a compact dynamical system (the Duffing equation). This canonical system plays the role of a ubiquitous model of non-stationary supervised classification problems. Second, we estimate the coherence of class-trajectories in empirically reconstructed space of system states. We show how this analysis can discern variations attributable to non-autonomous deterministic processes from stochastic fluctuations. The analyses are benchmarked using simulated and two different real datasets which have been shown to exhibit attractor dynamics. As an illustrative example, we focused on the analysis of the rat's frontal cortex ensemble dynamics during a decision-making task. Results suggest that, in line with recent hypotheses, rather than internal noise, it is the deterministic trend which most likely underlies the observed trial-to-trial variability. Thus, the empirical tool developed within this study potentially allows us to infer the source of variability in in-vivo neural recordings.


Text S2. Illustrative dataset in a non-stationary environment
This supplementary gives an illustration of the analytical results obtained in a well-known oscillatory multivariate time series where attractors subtly drift over time. Figure S2A shows the hourly ozone (O 3 ) ground concentration, nitric oxides (NO 2 , NO ) temperature and relative humidity over the period of a week. Ozone is an atmospheric pollutant synthesised from a series of precursor gases (mainly NO 2 ) by the catalysis of solar radiation.O 3 levels were grouped in three classes (low, moderate and high), and a discriminant defined in an optimally expanded phase space was then used to map precursors and atmospheric variables ( Figure S2A, bottom) to the three O 3 ranges.
Given the complexity of the atmospheric system dynamics, it is very difficult to discern the deterministic processes involved in the seasonal, non-stationarity trend observed in O 3 concentrations. We will address this issue empirically in a simple example with the analyses developed here.
Dataset. Elevated concentrations of tropospheric ozone are known to affect human health and vegetation adversely, thus time series of O 3 concentrations have been extensively studied for decades e.g. [1][2][3]. Ozone concentration is known to exhibit regular daily oscillations ( Figure S2A) and seasonal behaviour; with maximum concentrations and stronger periodicity in summer, becoming progressively weaker during the autumn weeks. Thus, this dataset is particularly useful as an illustration of our analyses. Data used in this research belongs to the Department of Agriculture, Generalitat Valenciana (Regional Government), Valencia, Spain; and it was recorded in a rural area of particular agricultural interest (39º 7' N; 0º 27' W). The station has continuously monitored ambient concentrations of pollutants (Dasubi 1008 RS, 2108, 8001) as well as meteorological variables mentioned above (Skye SKH 2013/1) since 1996 (8 weeks from August 12 were used here). Data was collected and averaged over 60 min intervals (for more details in data pre-processing, see [2]) Analyses. The original phase space was spanned by the hourly averaged concentrations of NO 2 , NO and hourly mean values temperature and relative humidity. Data was centred and normalized and no missing values were reported during the period used. Optimally regularized kernel discriminant operating on a third order expanded space (see methods) was used to map the ozone ranges to the phase space variables. Regularization penalty (here the optimal penalization constant was 4.8% of the mean kernel matrix, see [4] for details on the process) of the discriminant was optimized by cross validation on TD on a separate dataset. Classification subspace was then computed for the first trial (a week of hourly recordings) and held fixed for the following trials.
Results are shown in Figure S2B. Classification error and trajectory divergence increase monotonically until week 8, where TD increases abruptly and CE does not. This change occurs precisely at trial 8, when CE exceeds the confidence level at trial 1. To benchmark further the results, bootstraps (n=200) were constructed by shuffling the temporal order within trajectories (yet not shuffling across trials). As expected, bootstrap data do not show any trend. Figure S2B (inset) also shows an alternative measure to CE, namely the certainty of the classification which in our case is simply the normalized difference between the highest and the lowest discriminate z-score of all classes [5]. Interestingly, the increase in TD is indicated in this system by a sharp decrease in the average certainty.
In summary, as in Duffing dynamical system simulations, this result suggests a deterministic trend underlying the observed variability of ozone concentrations as would be expected from previous studies (e.g. [1][2][3]).